Geometric vector bundles/Locally free sheaves/Correspondence/Section

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Let denote a scheme. A scheme equipped with a morphism

is called a geometric vector bundle of rank over if there exists an open covering and -isomorphisms

such that for every open affine subset , the transition mappings

are linear automorphisms, i.e. they are induced by an automorphism of the polynomial ring given by .

Here we can restrict always to affine open coverings. If is separated then the intersection of two affine open subschemes is again affine and then it is enough to check the condition on the intersections. The trivial bundle of rank is the -dimensional affine space over , and locally every vector bundle looks like this. Many properties of an affine space are enjoyed by general vector bundles. For example, in the affine space we have the natural addition

and this carries over to a vector bundle, that is, we have an addition

The reason for this is that the isomorphisms occurring in the definition of a geometric vector bundle are linear, hence the addition on coming from an isomorphism with some affine space over is independent of the choosen isomorphism. For the same reason there is a unique closed subscheme of called the zero-section which is locally defined to be . Also, multiplication by a scalar, i.e. the mapping

carries over to a scalar multiplication

In particular, for every point the fiber is an affine space over .

For a geometric vector bundle and an open subset one sets

so this is the set of sections in over . This gives in fact for every scheme over a set-valued sheaf. Because of the observations just mentioned, these sections can also be added and multiplied by elements in the structure sheaf, and so we get for every vector bundle a locally free sheaf, which is free on the open subsets where the vector bundle is trivial.

A coherent -module on a scheme is called locally free of rank , if there exists an open covering and -module-isomorphisms for every


Vector bundles and locally free sheaves are essentially the same objects.

Let denote a scheme. Then the category of locally free sheaves on and the category of geometric vector bundles on are equivalent. A geometric vector bundle corresponds to the sheaf of its sections, and a locally free sheaf corresponds to the (relative) spectrum of the symmetric algebra of the dual module .

The free sheaf of rank corresponds to the affine space over .